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Now that you know about the prime factorization theorem, you can watch this video to learn about different classical techniques to split big numbers into primes. Specifically, this video talks about Pollard's rho method and Pollard's p-1 method. Pay attention to the examples in the video to see how these methods find the right primes. You can also try making your own examples to make sure you understand how these methods work.
Source: Richard E Borcherds, https://www.youtube.com/watch?v=Imxx1tFDeSw This work is licensed under a Creative Commons Attribution 3.0 License.